3. Consider the following utility optimization problem: max(min) U(r, y) (+2)(y1) subject to Pr P2y B (a) Write down the Lagrangian objective function and find the optimal values r, y, U* and A where A is the Lagrange multiplier (b) Find the derivative of U with respect to B, what do you notice? (c) How would you interpret A ? (d) Study the comparative static analysis with respect to B using two methods. Make sure that your answers from both methods are the same. (e) Under what conditions do the optimal values of r, y pertain to a maximum? ( Study the comparative static analysis with respect to P Now, consider the following minimization problem: Minimize B PxPy subject to (z 2)(y1)U where U* is the value function obtained in part (a). (g) Write down the Lagrangian objective function and find the optimal values r,y, B, and where u is the Lagrange multiplier. (h) Interpret (i) Under what conditions do the optimal values r, y pertains to a minimum? Analyse the comparative static analysis with respect to U and P (k) Using your answers to part (a), find the values of a, y, U, and A* if P 4, P26, and B 130